Due to the natural way that Bayesian inference deals with uncertainty quantification, several authors have recently proposed Bayesian analogues of \(L^2\)  calibration. [1]} noted that minimising the \(L^2\)  norm imposes orthogonality of the bias function and the gradient of the mathematical model. Under the [2]} framework, this orthogonality places constraints on the Gaussian process prior for the bias function. [3]} pointed out that orthogonality of the bias and mathematical model gradient is a necessary, but not sufficient condition, for minimising the \(L^2\)  norm. Instead, they assumed a prior distribution on the \(L^2\)  norm penalising “large" bias functions. [4]} proposed a different approach, not under the [2]} framework. They assumed a Gaussian process prior for the physical system, \(\mu (\mathbf {x})\) . The resulting posterior distribution for the \(L^2\)  norm induces a posterior distribution for \(\theta _{L^2}\) .
